A Directional Mixed Effects Model for Compositional Expenditure Data

Compositional data are vectors of proportions defined on the unit simplex and this type of constrained data occur frequently in Government surveys. It is also possible for the compositional data to be correlated due to the clustering or grouping of the observations within small domains or areas. We propose a new class of the mixed model for compositional data based on the Kent distribution for directional data, where the random effects also have Kent distributions. One useful property of the new directional mixed model is that the marginal mean direction has a closed form and is interpretable. The random effects enter the model in a multiplicative way via the product of a set of rotation matrices and the conditional mean direction is a random rotation of the marginal mean direction. In small area estimation settings, the mean proportions are usually of primary interest and these are shown to be simple functions of the marginal mean direction. For estimation, we apply a quasi-likelihood method which results in solving a new set of generalized estimating equations and these are shown to have low bias in typical situations. For inference, we use a nonparametric bootstrap method for clustered data which does not rely on estimates of the shape parameters (shape parameters are difficult to estimate in Kent models). We analyze data from the 2009–2010 Australian Household Expenditure Survey CURF (confidentialized unit record file). We predict the proportions of total weekly expenditure on food and housing costs for households in a chosen set of domains. The new approach is shown to be more tractable than the traditional approach based on the logratio transformation.